Characterization of T0 Space by Closed Sets
Jump to navigation
Jump to search
Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.
Then
- $T$ is a $T_0$ space if and only if
Proof
Sufficient Condition
Let $T$ be a $T_0$ space.
Let $x, y \in S$ such that
- $x \ne y$
By definition of $T_0$ space:
- $\left({\exists U \in \tau: x \in U \land y \notin U}\right) \lor \left({\exists U \in \tau: x \notin U \land y \in U}\right)$
WLOG: Suppose:
- $\exists U \in \tau: x \in U \land y \notin U$
By definition:
- $\complement_S\left({U}\right)$ is closed
where $\complement_S\left({U}\right)$ denotes the relative complement of $U$ in $S$.
By definition of relative complement:
- $x \notin \complement_S\left({U}\right) \land y \in \complement_S\left({U}\right)$
Thus:
- $\exists F \subseteq S: F$ is closed $\land\, x \notin F \land y \in F$
$\Box$
Necessary Condition
This statement follows mutatis mutandis.
$\blacksquare$
Sources
- Mizar article TSP_1:def 4